A hybrid stochastic Galerkin method for uncertainty quantification applied to a conservation law modelling a clarifier‐thickener unit

Abstract

The continuous sedimentation process in a clarifier-thickener can be described by a scalar nonlinear conservation law for the local solids volume fraction. The flux density function is discontinuous with respect to spatial position due to feed and discharge mechanisms. Typically, the feed flow cannot be given deterministically and efficient numerical simulation requires a concept for quantifying uncertainty. In this paper uncertainty quantification is expressed by a new hybrid stochastic Galerkin (HSG) method that extends the classical polynomial chaos approximation by multiresolution discretization in the stochastic space. The new approach leads to a deterministic hyperbolic system for a finite number of stochastic moments which is however partially decoupled and thus allows efficient parallelisation. The complexity of the problem is further reduced by stochastic adaptivity. For the approximate solution of the resulting high-dimensional system a finite volume scheme is introduced. Numerical experiments cover one-and two-dimensional situations.

Stochastic Galerkin framework with locally reduced bases for nonlinear two-phase transport in heterogeneous formations

“…This is quite different from the approach taken in previous work, e.g. [23,22], and therefore their hyperbolicity analysis cannot be extended to this case.…”

“…Sampling based generalized polynomial chaos methods, such as stochastic collocation [21] suffer from the curse of dimensionality, and they become infeasible for large problems due to the prohibitive computational cost. With the continuous growth of computer power, stochastic Galerkin methods including the efficient adaptive and parallelized hybrid stochastic Galerkin solver in [22], have been gaining popularity as a powerful alternative to sampling based methods.…”

A posteriori error analysis for random scalar conservation laws using the stochastic Galerkin method

“…Due to the relative entropy framework we expect our theory to be extendable to this case. For further applications of our method, the construction of space-stochastic adaptive schemes using the Hybrid Stochastic Galerkin method (Bürger et al (2014)) and the residuals as local indicators, will be considered.…”

“…For an overview on recent work on Uncertainty Quantification for hyperbolic equations see Bijl et al (2013); Le Maître & Knio (2010); Pettersson et al (2015). New numerical schemes for the SG system can be found in Jin & Ma (2017); Bürger et al (2014); Wan & Karniadakis (2006) and for the convergence analysis of approximate solutions of the SG system see Gottlieb & Xiu (2008); Hu et al (2015); Zhou & Tang (2012). However, in all these works the dimension of the discrete stochastic space is chosen in an ad hoc way, in particular independent of the spatial-temporal resolution.…”

A hyperbolicity-preserving discontinuous stochastic Galerkin scheme for uncertain hyperbolic systems of equations

“…Further developments of the Multi-Element approach encompass h-and hp-adaptive refinements in the stochastic space ( [33,34,36]) or a multi-resolution discretization using wavelets instead of gPC, cf. [6,24]. Another approach which ensures hyperbolicity of the resulting nonlinear SG system is the intrusive polynomial moment method.…”